Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
DOCUMENTOS DE TRABAJOThe Pass-Through of Large Cost Shocks in an Inflationary Economy
Fernando AlvarezAndy Neumeyer
N° 844 Octubre 2019BANCO CENTRAL DE CHILE
BANCO CENTRAL DE CHILE
CENTRAL BANK OF CHILE
La serie Documentos de Trabajo es una publicación del Banco Central de Chile que divulga los trabajos de investigación económica realizados por profesionales de esta institución o encargados por ella a terceros. El objetivo de la serie es aportar al debate temas relevantes y presentar nuevos enfoques en el análisis de los mismos. La difusión de los Documentos de Trabajo sólo intenta facilitar el intercambio de ideas y dar a conocer investigaciones, con carácter preliminar, para su discusión y comentarios.
La publicación de los Documentos de Trabajo no está sujeta a la aprobación previa de los miembros del Consejo del Banco Central de Chile. Tanto el contenido de los Documentos de Trabajo como también los análisis y conclusiones que de ellos se deriven, son de exclusiva responsabilidad de su o sus autores y no reflejan necesariamente la opinión del Banco Central de Chile o de sus Consejeros.
The Working Papers series of the Central Bank of Chile disseminates economic research conducted by Central Bank staff or third parties under the sponsorship of the Bank. The purpose of the series is to contribute to the discussion of relevant issues and develop new analytical or empirical approaches in their analyses. The only aim of the Working Papers is to disseminate preliminary research for its discussion and comments.
Publication of Working Papers is not subject to previous approval by the members of the Board of the Central Bank. The views and conclusions presented in the papers are exclusively those of the author(s) and do not necessarily reflect the position of the Central Bank of Chile or of the Board members.
Documentos de Trabajo del Banco Central de ChileWorking Papers of the Central Bank of Chile
Agustinas 1180, Santiago, ChileTeléfono: (56-2) 3882475; Fax: (56-2) 3882231
Documento de Trabajo
N° 844
Working Paper
N° 844
The Pass-Through of Large Cost Shocks in an Inflationary
Economy
Abstract
We describe several events in Argentina between 2012 and 2018 in which public utilities or the
exchange rate increase by a large amount in a single month. During these months the nominal
increase in our measure of the cost of the goods included in the core CPI was as high as 10%.
Motivated by these events, we use a menu cost model of price adjustment to derive the theoretical
pass-through of large cost shocks to consumer prices for an economy with an underlying inflation
rate of the order of 25% per year, as well as the impact of these shocks on the size and on the
frequency of price changes. Using a comprehensive, never used in the price adjustment literature,
micro-data set underlying the construction of the core CPI for the city of Buenos Aires we compare
the theoretical effect of the cost shocks to the empirical one. As predicted by the theory we find that
despite the high level of underlying inflation, the large frequency of both price increases and
decreases points to the importance of idiosyncratic firm shocks, such as the ones we use in our
model. Right after large increases in costs, both the fraction of price increases and the average size of
price increases dramatically rises, while the size of price decreases stays approximately the same,
just at the simple model predicts. On the other hand, the model predicts a small decrease in the
fraction of price decreases which we don't observe in the data. We conclude that for large sudden
changes in cost, consumer prices behave almost as if they were fully flexible. On the other hand, for
small shocks the short-term pass-through will be smaller and its half-life larger.
Resumen
Describimos varios eventos ocurridos en la Argentina entre 2012 y 2018 cuando los servicios
públicos o el tipo de cambio muestran grandes aumentos en un solo mes. Durante esos meses, el
incremento nominal según nuestra medición del costo de los bienes que componen la canasta básica
del IPC llegó hasta el 10%. Motivados por estos hechos, utilizamos un modelo de costo de menú de
ajuste de precios para derivar el traspaso teórico de grandes shocks de costos hacia los precios al
consumidor en una economía cuya tasa de inflación subyacente es del orden del 25% anual, y el
impacto de estos shocks en la magnitud y la frecuencia de las variaciones de precios. Utilizando un
conjunto integral de microdatos, nunca antes utilizado en la literatura de ajuste de precios, que
subyace a la construcción del IPC básico de la ciudad de Buenos Aires, comparamos los efectos
teórico y empírico de los shocks de costos. Tal como predice la teoría, encontramos que, a pesar del
alto nivel de inflación subyacente, la alta frecuencia tanto de las alzas como de las bajas de precios
apunta a la importancia de los shocks idiosincrásicos en la empresa, como los que utilizamos en
We thank participants at the XXII conference in Santiago de Chile, seminar at the Federal Reserve Banks of Minneapolis, St Louis and Chicago, and at the Mantel lecture at the AAEP meeting in Argentina. We especially thank the comments by David Lopez-Salido.
Fernando Alvarez
University of Chicago
and NBER
Andy Neumeyer
Universidad
Torcuato Di Tella
nuestro modelo. Inmediatamente después de un gran aumento de costos, tanto la fracción del alza de
precios como su magnitud promedio aumentan drásticamente, mientras que la magnitud de las bajas
de precios se mantiene casi igual, tal como predice el modelo simple. El modelo también predice una
pequeña disminución en la fracción de las bajas de precios que no observamos en los datos.
Concluimos que, ante un gran cambio repentino en los costos, los precios al consumidor se
comportan casi como si fueran totalmente flexibles. Por otro lado, si el shock es pequeño, el traspaso
a corto plazo será menor y su duración será mayor.
1 Introduction
This paper surveys and modestly extends the theory of menu cost models for the behavior
of the aggregate price level after large cost shocks. It does so in the context of an economy
with a high underlying rate of inflation. It concentrates on the effect of large permanent
unexpected increases in the nominal price of inputs on the price level at different horizons.
We use a simple theoretical model where increases in nominal cost will increase aggregate
prices one for one in the long run. We study how the nominal rigidities implied by a menu
cost distribute the increases in the price level between the impact effect immediately after
the cost shock and the subsequent price adjustment until the price catches up with its long
run increase. In other words, we study the pass-through of large cost shocks at different
horizons. We pay particular attention to the role of the underlying inflation rate as well as
to the role of the size of the cost shock, since both elements are important to determine the
dynamics of aggregate prices.
Our interest in that question comes both from interest in testing aspects of price setting
theories and from a practical monetary policy point of view. On the theoretical side, the
differential effect of large versus small shocks is the hallmark difference between menu cost
models and time dependent models of price adjustment. Hence, the characterization of
the model’s behavior is important to be able to discern between theories with observed
experiences. On the monetary policy side, we are interested in this particular question
because of the recent experience in Argentina, an economy where inflation has been quite high
for international standards in the last decade, and where due to changes in macroeconomic
policies in the last four years there have been large cost shocks. In particular, there have been
large changes in exchange rates as well as extremely large changes in the price of regulated
prices, mainly inputs related to energy. In this context we ask the question of whether the
response to these large cost shocks is close to one of an economy with fully flexible prices or
to one with time dependent price setting, such as the Calvo model.
We give a short narrative of instances where in a single month there have been very
1
large cost changes in the inputs for goods that make up the core CPI for Argentina between
2012 and 2018. We also use a comprehensive, never used in the price adjustment literature,
micro-data set underlying the construction of the core CPI for the city of Buenos Aires to
compute the objects analogous to the ones we describe in the theory. From the comparison
between statistics in the model and in the data we conclude that either full price flexibility
or time dependent rules are quite counterfactual. Nevertheless, in the cases of very large
positive cost shocks in an economy with an underlying high inflation, the pass-through of
the shock to prices is quite fast. In the most extreme case, when the exchange rate jumped
25% (from 38.73 pesos per dollar to 30.95) in the last three days of August, the fraction of
firms changing prices which is 24% in normal times rose to almost 60% between August and
September 2018.
We believe that through this narrative approach we can make a plausible case that reg-
ulated prices can be thought as exogenous changes in cost, and that the same will be true
for some of the large changes in exchange rates. The change in regulated prices followed a
period of many years during which despite the fact that inflation was very high for inter-
national standards, regulated prices were frozen in nominal terms. The situation became
untenable both from the fiscal point of view, costing at least 3% of GDP, and also due to the
distortions that this policy generated. The normalization of prices occurred in steps due to
the very large increases that it implied, as well as due to challenges in the courts. To get an
idea of the magnitude of the change in relative prices, during the sample period the price of
natural gas relative to the core CPI was mutiplied by a factor of five and the relative price
of electricity by a factor of three. On the exchange rate changes there were different reasons
for the observed large changes. In 2014 there was a devaluation within a regime with severe
capital controls and dual exchange rates. At the end of 2015 and beginning of 2016 there
was a large devaluation when the multiple exchange rate system was abandoned essentially
overnight. The sharp depreciation of the exchange rate throughout the period April-August
2018 is probably the result of a mix of a reaction to a change in economic policies (perhaps
2
larger than what policymakers anticipated) and of exogenous shocks within the framework
of a dirty floating exchange rate regime. In our model cost shocks are once and for all
unexpected changes, which may not be completely accurate for some of the episodes.1
We use the micro data underlying the construction of the consumer price index (CPI) in
the city of Buenos Aires to compare the predictions of the model with actual observations after
several of the large cost increases that occurred between 2014 and 2018. Some readers may
be aware that the national statistical agency produced price indices that are widely regarded
as underestimating the inflation rate between 2008 and 2015, see for instance Cavallo (2013).
The price index of the city of Buenos Aires has the advantage that it was independent of the
national statistical agency and produced figures very similar to the privately produced price
indexes.
We propose a theoretical model where firms have both idiosyncratic as well as aggregate
changes in costs. We assume that firms are monopolistic competitors and that they face a
fixed menu cost for changing prices. The solution to the firms’ price setting problem gives
rise to a classical sS rule for price changes, which postulates price increases as well as price
decreases. The optimal decision rule, as well as the response of the aggregate price level
depend crucially on the ratio of the inflation rate to the variance of the idiosyncratic shocks.
While idiosyncratic shocks make the analysis more complicated, we think that they are
essential to the answer of the problem for two reasons. First, they are required to reproduce
the large fraction of price decreases that are observed even when inflation rate is above
25% per year. Second, the behavior of the pass-through depends on the magnitude of these
shocks relative to the level of inflation. In the theoretical section we compare the effect of
cost increases in three cases: the case of very low inflation rate, which is the case of most
economies, high inflation rates of the order of Argentina during this period (say 25% per
1For instance, the normalization of the exchange rate system and removal of capital controls is a policythat was to a large extent announced by the two main parties before the election at the end of 2015. Indeedthere is an unsettled debate in the Argentine economic circles on whether the effect of the likely increase inthe exchange rate that the unification of the exchange rate markets will entail was anticipated and includedin price changes months before it happened. We discuss these episodes in more detail in section 2.
3
year), and very high inflation. The degree of pass-through depends on both the size of the
steady state inflation rate as well as the size of the shocks. It turns out that even for inflation
rates as large as 25% per year, still one can see clear effects of price stickiness. Yet for large
cost increases, of the magnitude that occurred in Argentina, the pass-through occurs in an
extremely short time period. In particular, there is a very large impact effect, so most of the
adjustment occurring at the time of the shock, and a very short half life, smaller than two
months, for the remaining adjustment. Indeed this matches what we see during the months
of large cost increases in Argentina: a very sharp increase in the fraction and in the size of
price increases, almost no change in the fraction and in the size of price decreases, and a
jump on the inflation rate close to the size of the cost increase.
It is well known that cost shocks explain a large fraction of the variance of inflation
in standard estimates of medium size new-keynesian models. Nevertheless, these costs are
typically a residual in the standard specification. On the other hand, there is a literature
that tries to use identified cost shocks to evaluate different price setting mechanism. Our
paper adds to the literature that studies the effect of large cost shocks in price setting models
with menu costs of price adjustment. Early examples of this literature are Gagnon (2009)
and Gagnon, Lopez-Salido, and Vincent (2013). These papers consider the experience of
Mexico in the mid 90’s where there was both a large step devaluation (about 40%) as well
as changes in the VAT (from 10% to 15%). Another similar recent exercise is the one in
Karadi and Reiff (2018), which uses changes in the VAT in Hungary. In both cases, a version
of a menu cost model is used to interpret the micro data underlying the construction of
CPI and to compare the predictions of this class of models with the data. A related, yet
different evidence is the study of the exchange rate pass-through to export and import prices
in customs data in Bonadio, Fischer, and Saur (2016) comparing small changes in the Swiss
franc’s value with the large change that occurred when the Swiss National Bank abandoned
its ”peg” to the euro in January 2015. Finally, Alvarez, Lippi, and Passadore (2016) estimates
panel regressions of the short term pass-through of exchange rate changes to consumer prices
4
that include non-linear terms in the size of the exchange rate changes. Differently from
the previous studies, these panel regressions do not use micro price statistics. Relative to
Gagnon (2009), Gagnon, Lopez-Salido, and Vincent (2013), and Karadi and Reiff (2018),
and motivated by the levels of inflation in Argentina during the period of interest, this paper
has a more systematic treatment of the role of the running (or steady state) inflation and of
the size of the cost shocks on the level of short term pass-through, and on the overall speed
of adjustment. Finally, relative to Alvarez et al. (2019) we study a different period of time
for Argentina, but more importantly, in this paper we concentrate on the effect that large
unexpected cost shocks have on price dynamics, as opposed to the effect of different steady
state inflation levels which is the main point of Alvarez et al. (2019).
The remainder of the paper is organized as follows. In section 2 we provide a brief
narrative of macroeconomic events in Argentina in the period 2012-2018 as they pertain to
nominal cost shocks and inflation. The theoretical analysis is in section 3. The firm’s problem
is described in section 3.1, section 3.2 describes the steady state distribution of price markups
over nominal marginal costs, section 3.3 explains how inflation affects optimal decision rules,
section 3.4 derives analytically expressions for the impulse responses of consumer prices to
cost shocks and section 3.5 contains numerical simulations of how cost shocks affect consumer
price dynamics for small and for large cost shocks, and for three different inflation rates (low,
high, and extremely high). Finally, section 4 compares the predictions of the model with the
evidence emerging from city of Buenos Aires consumer price micro-data.
2 Nominal Cost Shocks and Inflation, Argentina 2012-
2018
In this section we provide some background on the evolution of inflation and the nominal
value of some key inputs during our sample period, July 2012 to September 2018. We first
comment on the monetary policy framework and exchange rate developments and later on
regulated price policies for energy inputs.
5
We divide the analysis of the monetary policy framework in three periods: the dual
exchange rate regime prior to December 2015, the inflation targeting regime between March
2016 and December 2017, and the abandonment of the inflation targeting regime throughout
2018.
During the first period the monetary policy framework was a dual exchange rate regime.
The central bank fixed the Argentine peso price of the US dollar for transaction related to
international trade as well as for some limited financial ones. Capital controls were binding
and a shadow exchange rate market that carried a premium over the official one developed.
We believe that this was caused by the high rate of money growth due to the monetary
financing of deficits. Part of the monetary financing of deficits was sterilized with central
bank debt that reached close to 6% of GDP in March 2016. The average rate of core inflation
between July 2012 and December 2015 in the City of Buenos Aires was 31% (with a median
of 28%).2 The official exchange rate crawled at a median rate of 17% (annualized median
monthly rate of devaluation). Between November 2013 and March 2014 a succession of jumps
in the exchange rate resulted in a cumulative increase of 32% in the exchange rate. Core CPI
prices in those months rose by 15%.
On December 15th, 2015 exchange rate controls were removed and the exchange rate
started to float with limited central bank intervention. The unification of the foreign exchange
market entailed a depreciation of the peso of over 50% between December 2015 and February
2016 (see table 1 and figure 1). In March 2016 the central bank adopted an interest rate
peg as its policy instrument in the context of an incipient inflation targeting regime which
was formally adopted in September. The national inflation targets were 12–17% for 2017,
10% ± 2% for 2018, and 5% ± 1.5% for 2019. The actual core inflation rates in the city of
Buenos Aires were 35% in 2016, 24% in 2017, and 43% in 2018. Wages increased 34% in
2016, 24% in 2017, and 21% in the first nine months of 2018 (prices increased by 30% in this
period).3
2Average and median of annualized monthly inflation.3Core inflation for the city of Buenos Aires. Wages are from Ministerio de Trabajo (2018)
6
Starting in December of 2017 there was a gradual abandonment of the inflation targeting
framework. On December 28th 2017 the inflation target for 2018 was raised from 10% to
15% and in January the central bank lowered interest rates by 150 basis points from 28.75%
to 27.25%. The market perception was that this interest rate move was motivated by po-
litical pressure. A speculative run on the central bank’s debt unraveled between April and
September 2018. In the period between April and September (end of period) the central
bank paid (did not rollover) 529 billion pesos of short term debt (40% of its debt and 53%
of the monetary base in April) and the monetary base increased by 250 billion pesos in the
same period (see Bassetto and Phelan (2015) for a related theoretical model).4 This led to a
sharp depreciation of the Argentine peso. The peso-dollar exchange rate quivered around 17
pesos per dollar in the period July-December 2017, then around 20 pesos per dollar between
late January and the end of April, and reached reached 41 pesos per dollar in late September.
The evolution of the (log) exchange rate is depicted in figure 1.
Public utilities prices were also an important source of large nominal cost shocks during
our sample period. The nominal price of regulated goods such as electrical power, natural
gas, water, and public transportation was practically frozen for a decade up to 2014. With
the price level rising, the relative price of regulated goods, especially energy items, steadily
eroded. The gap between the nominal marginal cost of these regulated goods and their
sale prices was covered with government subsidies that in 2015 are conservatively estimated
around 3% of GDP. A small attempt at normalizing these prices resulted in large nominal
increases in the price of natural gas during 2014. A more ambitious normalization started
2016.
Figure 1 provides an overview of the evolution of the nominal variables described in
the preceding paragraphs. All prices are from the City of Buenos Aires CPI.5 The solid
4Establishing the reasons behind this run is difficult and goes beyond the purpose of this paper. Severalexplanations have been proposed in local economic circles ranging from the importance of current accountdeficits, the effect of negative productivity shocks on agriculture (a very large drought), the size and maturityof the central bank debt, changes in local taxation of capital flows, the fear of fiscal dominance in the nearfuture, and the adverse international financial circumstances, to cite few of them.
5We work with data for the City of Buenos Aires as the reliability of national statistics between 2007 and
7
lines depict the natural logarithm of the core CPI, regulated prices in the CPI, the price of
electricity, the price of natural gas, and the exchange rate. Core inflation excludes seasonal
products and regulated prices. All five are normalized to their July 2012 value.
Figure 1: Cost shocks and Inflation
Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan190
0.5
1
1.5
2
2.5
3
3.5
Logarith
mic
poin
ts
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Month
ly L
ogarith
mic
diffe
rence
Core
Regulated
Electricity
Gas
Exchange Rate
Core inflation
Note: Prices on the left axis are the natural logarithm of the price index published by theCity of Buenos Aires normalized . The exchange rate is the natural logarithm of the monthlyaverage published by the central bank. Core inflation is the monthly log difference of thecore price level (resto) which, excludes seasonaland regulated goods and services.
The black solid line is the price level for core goods and services while the doted black
line represents its rate of change (in log differences). The blue line represents the exchange
rate expressed as pesos per dollar. There are three major devaluations. During the dual
exchange rate period the exchange rate exhibits a rate of growth below that of core prices
2016 has been severely questioned. For example, on February 2013 the International Monetary Fund issueda declaration of censure against Argentina in connection with the inaccuracy of CPI data from the nationalstatistical office (INDEC). Also see Cavallo (2013) for a comparison of national statistics and online pricesacross several Latin American countries, including Argentina.
8
except for the period of the devaluation around January of 2014. There is a second sharp
depreciation of the peso between December of 2015 and February 2016 after the removal of
capital controls. The last episode of depreciation of the peso, about 80%, occurred between
April and September of 2018.
Regulated consumer prices in the city of Buenos Aires are represented by the green line
in figure 1. As it was the case for the exchange rate, prior to December 2015 these prices
grow at a slower pace than core prices and there are two important relative price corrections,
one in 2014 and another one in 2016. The main drivers of regulated prices are the prices of
energy, especially electricity and gas. The red line represents the price of electricity. It is a
step function with jumps of 253% in February 2016, 92% in January–March 2017, 68% in
December–February 2018 and 25% in August 2018. The price of natural gas follows a similar
pattern (see table 1).6
We summarize the behavior of nominal cost shocks in a proxy variable the we refer to as
cost proxy of cost shock. We assume that consumer goods are produced with labor, tradable
inputs, and regulated goods. In our model we assume that there is a consolidated sector
that produces and retails consumption goods. It purchases an aggregate input in a flexible
price competitive market and sells its output to consumers in a monopolistically competitive
market. To construct our intermediate input measure, we assume following Jones (2011)that
the share of intermediate inputs in the production of the final consumption goods is 50%, and
that the remaining is a labor share of 50%. The weights of energy and tradable intermediate
goods are 10% and 40%, respectively. Hence, our measure of cost shocks is obtained by
computing first a geometric price index for this aggregate input as p0.1R E0.4W 0.5, where pR
denotes regulated prices, E is the peso/US dollar exchange rate, and W are nominal wages.
Figure 2 depicts the cost shock proxy variable, wholesale prices and core inflation.7 The proxy
6The price of natural gas fell 68% in August 2016 because the Supreme Court stayed the April priceincrease on procedural grounds. The price increases were resumed after the executive remedied the judicialobjection to the previous price increase.
7There is no official reliable data on wholesale prices prior to December 2015.
9
Table 1: Large Nominal Cost Shocks
Electric- Natural Exchange Electric- Natural Exchangeity Gas Rate ity Gas Rate
Oct-12 9 Feb-17 48Nov-12 15 18 Mar-17 30Jan-14 12 Apr-17 31Feb-14 11 Jul-17 7Apr-14 57 Dec-17 43 42Jun-14 32 Jan-18 8Aug-14 29 Feb-18 17Dec-15 19 Apr-18 36Jan-16 19 May-18 17Feb-16 253 8 Jun-18 12Apr-16 184 Aug-18 25 9Jul-16 5 Sep-18 28Aug-16 -68 Oct-18 27Oct-16 134 Nov-18 6Nov-16 14
Note: We report the percentage change between prices in month t and t − 1. Price data forelectricity and gas is from the Consumer Price Index for the City of Buenos Aires. Exchange rateis the change in the monthly average exchange rate.
10
for nominal cost shocks tracks closely wholesale prices with a correlation of 0.92. Observe
that spikes in nominal cost shock inflation are associated with spikes in core inflation.
Figure 2: Cost shocks proxy and wholesale prices
Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan190
0.05
0.1
0.15
Month
ly logarith
mic
diffe
rence
Wholesale inflation
Cost
Consumer core inflation
Note: All variables are monthly log differences. Cost = p0.1R E0.4W 0.5. Theexchange rate is the log difference of the monthly average. Consumer pricesare from the City of Buenos Aires Statistical Office. Wholesale prices (IPIM)from INDEC. Exchange rates from the central bank. Wages from Ministe-rio de Trabajo (2018). Regulated prices from the sub-component of theCPI.
In the next section we present a theoretical model of the pricing decision of a retail firm
in order to study the speed and the magnitude of the pass-through of these cost shocks to
consumer prices.
3 Pass-through of cost shocks to prices: theory
In this section we study the nominal pass-through to consumer prices of the nominal cost
shocks in a menu cost model of price adjustment. We consider the problem of a monop-
olistically competitive firm that faces idiosyncratic demand and productivity shocks in an
11
environment in which the cost of an aggregate input grows at a constant inflation rate. We
study how the aggregate consumer price level reacts to an unexpected jump in nominal
marginal costs in this environment.
We first describe the firm’s price setting problem in section 3.1 and then look at the cross
sectional distribution of prices in section 3.2. We then proceed to describe how steady state
inflation affects decision rules in section 3.3 that refers to Alvarez et al. (2019). Finally,
in section 3.4 we report the impulse response of price distributions to cost shocks and in
section 3.5 we compare the reaction of prices to small and large cost shocks and find interesting
nonlinear effects.
3.1 Price setting problem for the firm
We consider an economy where firms’ marginal nominal cost have a common and an idiosyn-
cratic component. The common component is given by a nominal cost, which we will assume
that after an initial value is realized will grow at a deterministic constant rate. The (log of
the) idiosyncratic component follows driftless Brownian motion, with innovation variance σ2.
Firms face a downward slopping demand, and act as monopolistic competitors and must pay
a fixed menu cost to adjust their price. The firm’s marginal (and average) cost will then be
W (t)x(t), where W (t) is the nominal cost of the aggregate input and x(t) is an idiosyncratic
shock. We assume that x(t) = exp (σB(t)), where B(t) is a standard brownian motion, inde-
pendent across firms. We will start the firms at a steady state, where Wt has been growing
a constant inflation rate π, so that W (t0 + T ) = W (t0)eTπ.
We will use g to describe the logarithmic deviation of the firm’s current markup relative
to the one that maximizes instantaneous profits. We refer to this variable as the “price gap”.
The price gap is positive, g > 0, if the price of the product is relatively high or if the firm’s
cost is relatively low. We will assume that the optimal markup is independent of the level of
the costs and of the demand, as it will be the case with an iso-elastic demand function and
constant marginal cost. Note that during a period where the firm does not change prices,
12
its markup changes only because it cost changes, i.e. dg(t) = −d logW (t) − d log x(t). For
instance, at steady state we have that dg = −πdt−σdB during the period at which the price
of the good does not change. At the times when the firm decides to pay the fixed cost and
change prices g(t) changes discretely and in equal proportion (equal in log points) than the
change in price.
The firm problem can be summarized by the Bellman equation:
V (g) = maxτ
E[∫ τ
0
e−ρtF (g(t)) dt+ e−ρτ[ψ + max
gV (g)
]| g(0) = g
](1)
subject to dg(t) = −πdt− σdB(t) for all 0 ≤ t ≤ τ and g(0) = g (2)
In this problem ψ is the fixed cost, ρ is the (real) discount rate, π is the inflation rate
(of the aggregate input), σ is the idiosyncratic volatility of the cost shocks, and F (·) is
the instantaneous profit function, written as a function of the price gap. The expectation
in the objective function is with respect to the cost shocks, the values of the path for B(t).
Mathematically speaking the object of choice τ are stopping times. A stopping times indicates
the time and circumstances under which prices will be adjusted. In this kind of problem
the optimal rule is that τ occurs the first time that g(t) is outside a range of inaction.
After paying the cost and deciding the optimal price we find the optimal return point as
g∗ = arg maxg V (g), or V (g∗) = maxg V (g). In Appendix B we write the o.d.e. and
boundary conditions which simultaneously determines the function V (·) and the values of
g, g∗, g. In Appendix C we given an alternative way to formulate and compute a discrete
time and discrete state problem that approximates it.
The optimal policy for the firm is an sS rule described by three numbers g, g∗ and g. We
refer to g < g as the boundaries of the range of inaction, and to g∗ ∈ (g, g) as the optimal
return point. The inaction region is thus described by the interval [g, g]. If the price gap g
is in the inaction region, the price of the firm stays constant and the price gap changes with
the cost changes—with opposite sign. If the price gap is lower than g, so that the markup is
13
very low, then the firm will pay the fixed cost ψ and increase prices so that right after the
price change the price gap will be g∗. Likewise, if the price gap is higher or equal than g,
so that the markup is very high, the firm will pay the fixed cost ψ and decrease its price, so
that right after the price change the price gap becomes g∗.
Note that we are writing the profit function F as a function of the price gaps exclusively.
In general, even at steady state, it should be a function of both the price and the cost, so
that profit equal markup times quantity demanded, which itself depends on relative prices.
We explain below the conditions under which this simplification can be obtained. We choose
units so that F is measured in either terms of units of the aggregate input, or as deviations
of maximized steady state profits –our preferred choice. Of course, the fixed cost ψ has to
be measured in the same units as F . The optimal decision rule depends only on the ratio
ψ/B, since the value function is homogeneous of degree one in ψ and B. Intuitively, the
fixed cost matters only through the relative advantages of changing prices, captured by the
curvature of F , rather than on each of them separately. Furthermore, the discount factor ρ is
a real interest rate, measuring the intertemporal price of the aggregate input. The inflation
rate π is also the nominal change in the price of the aggregate input. Since ρ, π and σ2 are
rates per unit of times, the optimal decision rule depends, besides on ψ/B, only on the ratios
{ρ/σ2, π/σ2}.
Derivation of the profit function F . We have written the profit function F having the
price gap g as its only argument. This is a simplification which provides lots of tractability.
In Alvarez et al. (2019) we work without this simplifying assumption, obtaining essentially
the same results.8 We describe here the assumptions so that we can use the simplified version.
Let Q(p/W )z be the quantity demanded as a function of p/W , the ratio of the nominal price
of the good p and the nominal price of the generic input W . In steady state we can write
this relative price or the relative price with respect to some aggregate good without loss of
8See the section on the model with random walk shocks and CES demands, which we refer to Kehoe andMidrigan (2015) version of the Golosov and Lucas’s (2007) model.
14
generality. It also turns out that in the set-up described by Golosov and Lucas (2007), which
is in no way pathological, this is a consequence of the general equilibrium structure. The
variable z is a multiplicative shifter of the demand, which we use for two different illustrations.
The nominal marginal cost is xW . We will proceed by steps. First we will derive the profit
function F with (g, z, x) as arguments. Then we will add assumptions to eliminate (x, z)
from it.
Let’s use m for the log of the optimal markup, i.e. let the nominal price that maximize
instantaneous profits be: P ∗ = em xW . Thus g is defined as
g ≡ log
(P/(Wx)
P ∗/(Wx)
)= log
(P
Wx/M
)= log
(P
Wx
)−m or P = Wxeg+m (3)
Now we are ready to write the profit function. The units of profits will be first in term of
the real value of the aggregate input. Profits in nominal terms are [P − xW ]Q (P/W, z), i.e.
nominal markup times quantity. Dividing this expression by W we obtain profits in time t
units of the aggregate input.
F (g, x, z) =
[P
xW− 1
]Q
(P
Wxx
)z x =
[eg+m − 1
]Q(eg+mx
)z x (4)
Furthermore, assuming that Q is iso-elastic, with elasticity equal to η, so that Q(P/W ) =
A (P/W )ηz for some constant A, then the optimal markup M , or its log m, will be constant
and equal to m = log(η/(η − 1)). In this case profits will be:
F (g, x, z) =[eg+m − 1
]e−η(g+m)Ax1−η z
It should be clear that, by definition, F is maximized when g = 0.
Adding the extra assumption that the demand shifter satisfy z = xη−1, then we obtain
that F does not depend on (x, z). To be honest, this is a strange assumption; it requires the
shock that increase cost shock to simultaneously push the demand up, so that the maximized
15
profit remains the same for any value of x. On the other hand it simplifies the algebra a lot!
As mentioned above, in Alvarez et al. (2019) we work out both version of the models finding
very small differences.
An alternative is to use a second order approximation of the function F (g, x, z) and to
retain only the leading terms on g. In particular, we use a second order approximation of
F (g, x, z) around around g = 0, x = x and z = z. Using that g = 0 maximizes profits, and
the multiplicative separable nature of the profit function in three terms, ignoring the terms
that are higher than second order, and the terms not involving g we have
F (g) ≡ −(η − 1)η
2g2 = −Bg2 so that B ≡ η(η − 1)/2 (5)
where we are measuring profits relative to the maximized profit at x = x and z = z, which
equals F (0, x) =(
ηη−1
)−η [Ax1−η zη−1
]. Of course, we can combine these assumptions too.
Lack of First order strategic complementarity. Finally, the result in equation (5)
is useful because it states that in the model we focus on, there is no first order strategic
complementarity. This is because we can summarize the behavoiur of the rest of the firms
in z. This will be the case in the standard neokeynesian model, where there will be at least
two effects on the profit function coming from aggregate consumption: one is that with CES
demand for each firm, higher output of each of the other firms shifts the demand up, and the
other, in opposite sign, that higher output decreases the Arrow-Debreu price of the good in
the current period. But as we have seen, these effects –captured by z– are of third order in
the profit function in the current set-up.9
Interestingly, this means that we can use as an accurate approximation the firm’s decision
rules characterized by {g, g∗, g} even if the rest of the economy is not at an steady state,
as long as the firm expects constant growth rate of the aggregate input prices. We will rely
heavily in this property to characterize the impulse response of prices to a one-time common
9The general equilibrium version in Golosov and Lucas (2007) also makes the value of the nominal aggre-gate input, labor in their case, depending only on the path of nominal money.
16
shocks to their cost.
3.2 Steady state distribution of price gaps and price changes
We let f(g) be the steady state density of the distribution of price gaps. This density has
support [g, g]. It solves a simple ordinary differential equation, balancing the flows in and
out when a price change occurs. Its shape depends on the parameters {g, g∗, g, π/σ2}. Of
course, g, g∗, g depend on all the parameters that define the steady state problem of the
firm described above, namely {ψ/B, ρ/σ2, π/σ2}, or using the simplification described in
Appendix B it is described by just {ψ/B, π/σ2}. The equations that determine the steady
state density given the decision rules and parameters are:
0 = πf ′(g) +σ2
2f ′′(g) for all g ∈ [g, g∗) ∪ (g∗, g] with boundary conditions:
0 = f(g) = f(g) and
∫ g
g
f(g)dg = 1 (6)
The first line is the Kolmogorov forward equation, and the second line has the three relevant
boundary conditions. The solution is given by the sum of two exponential functions. The
boundary conditions on the extreme of the inaction regions indicate that there is zero density
in those points. Intuitively, this is because they are exit points, so it is “hard” to accumulate
density near them. This result will be important for some of the results below.
The shape of the density f depends on the inflation rate relative to the idiosyncratic
variance π/σ2, both through their direct effect, as seen in equation (6) and indirectly through
their effect on g, g∗, and g. For finite π/σ2, the distribution has zero density in its two
boundaries, it is strictly increasing in (g, g∗), non-differentiable at g∗, and strictly decreasing
in (g∗, g). In particular for π/σ2 = 0 the distribution is symmetric around g∗ = 0, and
has a tent-shape, with f ′′(g) = 0. As π/σ2 increases, the value of g∗ becomes positive, and
the shape of f becomes concave from [g, g∗) and convex between (g∗, g]. As π/σ2 → ∞,
the distribution converges to a uniform distribution between [g, g∗] and to a zero density
17
everywhere else, as in the model of Sheshinski and Weiss (1979), which has σ2 = 0 and finite
π.
The change on the shape of the invariant distribution f reflects, in a very intuitive way,
the different strength of the idiosyncratic shocks (measured by σ2), which are symmetric, and
the effect of inflation (measured by π), which is asymmetric. As inflation increases, the price
gaps g naturally tend to piled up in the left side, since the cost increase in expected value
as the nominal price remains fixed. Indeed, as π/σ2 →∞ this effect is so strong, that price
gaps essentially march deterministically from g∗ to g, and hence the distribution is uniform,
as stated above. Lastly, the drastic change in shape of f around g∗ is also intuitive, since
after a price is changed the new price of the product is set so that g = g∗, which explains
why the mass is highest at this point, and why it is not differentiable since the behaviour of f
around this point is governed by mass coming from the boundaries of the range of innaction.
In steady state the size of price changes is given by a very simple formula of the thresholds
of the sS rule. Price increases are given by g∗− g, since the firm increases its price when the
markup is very low, i.e. the first time g reaches g. Likewise, price decreases occur when the
markup has reached the value g, and are of size g − g∗. Denoting the size price increases by
∆+p and the size price decreases by ∆−p we have that: ∆+
p = g∗ − g and ∆−p = g − g∗.
We denote the average number of price changes per unit of time by λa. This can be easily
computed as the reciprocal of the time between price changes, by the fundamental theorem
of renewal theory. The expected time until a price change T (g) for a firm with current price
gap g solves the following Kolmogorov backward equation:
0 = −πT ′(g) +σ2
2T ′′(g) for all g ∈ [g, g] with boundary conditions: 0 = T (g) = T (g)
The boundary conditions are quite natural, at either g or g there will be a price change,
and hence the expected time to reach them is zero! Since right after a price change g = g∗,
then the expected time until the next price change is T (g∗), and hence the frequency of price
changes λa = 1/T (g∗). Using a similar procedure we can find the frequency of price increases
18
λ+a and the frequency of price decreases λ−a . For instance, letting T +(g) be the expected
time until a price increase, it is easy to see that it satisfies the same Kolmogorov backward
equation than T . The difference is in the boundary conditions, which are T +(g) = T +(g∗),
since a price decrease will occur at g but we need to keep counting; we also have T +(g) = 0,
since a price increase occurs at g . The frequency of price increases is thus: λ+a = 1/T +(g∗).
A similar argument holds for the expected time until a price decrease T −, i.e. it solves
the same o.d.e. than T with boundary conditions T −(g) = T −(g∗) and T −(g) = 0. The
frequency of price decreases is thus: λ−a = 1/T −(g∗).
We note that the length of the range of inaction equals the sum of the average size of price
increases plus the average size of price decreases: g − g = ∆+p + ∆−p . This observation will
become useful because these average sizes can be measured, and the length of the range of
inaction is important to understand when a cost shock is large. Finally, the second moment
of price changes is given by E[∆2p] = λ+a
λa
(∆+p
)2+ λ−a
λa
(∆−p)2
.
In Appendix C we give a simple alternative numerical procedure using a discrete time
and discrete state version of the model to compute the steady state distribution and the
frequency of price changes per unit of time.
3.3 Optimal decision rules and inflation
This subsection analyzes how optimal pricing decisions vary with the rate of normalized
inflation, π/σ, keeping the normalized fixed cost, ψ/B, constant.10
For π/σ2 ≈ 0, then the decision rules are approximately symmetric, with g∗ = 0 and
g = −g. 11 This implies that around zero inflation the price increases and price decreases
have approximately the same size ∆+p = ∆−p . Moreover, the frequency of price increases and
price decreases is the same around zero inflation, so λ+a = λ−a .
For higher inflation rates relative to idiosyncratic volatility π/σ2, the optimal return
10In Appendix B shows that, fixing the fixed cost relative to the curvature of profits ψ/B, the optimaldecision rules depend on the normalized level of inflation π/σ2, as discussed in Appendix B.
11We say approximately because under the quadratic approximation at exactly π/σ2 = 0 the decision rulesare symmetric.
19
markup g∗ becomes larger. This is because, due to inflation, during inaction markups decrease
in expected value, and thus this expected effect is compensated by starting with a higher
markup. In Alvarez et al. (2019) we show that for small inflation the main adjustment is
not in the frequency of price changes λa, but instead in the difference between the frequency
of price increases and decreases, i.e. the derivative of λa(π) with respect to inflation is
zero at π/σ2 = 0, but the derivative of λ+a (π) − λ−a (π) is strictly positive. We also show
analytically that it accounts for 90% of the change in inflation at low inflation, the other
10% is explained by changes in ∆+p −∆−p . Summarizing, as we move from zero steady state
inflation (where frequency and size of price increases and decreases are symmetric) to positive
steady state inflation, the model predicts that the frequency of price increases λ+a (π) will be
higher than the one of decreases λ−a (π), and that the size of price increases and decreases
will be approximately the same, i.e. ∆+p ≈ ∆−p . Nevertheless, while while we show that
the average size is similar, we also show that as we move from zero steady state inflation,
∆+p > ∆−p .12
For very large inflation, λ+a (π)→ λa(π) and λ−a (π)→ 0, as π →∞, so most price changes
are increases, and the model converges to Sheshinski and Weiss (1979), in the sense made
precise in Alvarez et al. (2019).
3.4 Impulse response of the price level to unexpected cost shocks
In this section we characterize the impulse response of the aggregate (log of the) price level
to a once and for all increase in the nominal price of the aggregate input of size δ, measured
in logs. We will consider different inflation levels π and different sizes of the shock δ.
We start with an economy that is in the steady state distribution of price gaps. This
economy is characterized by parameters {ψ/B, ρ, σ2} and π. Firms will face a once and for
all jump on the nominal price of the aggregate input, and believe that after this jump the
price of the inputs will rise at the same inflation rate π as before the jump. We are interested
12For a precise statement see Propositions 1 and 3 in Alvarez et al. (2019). Numerically we find the changesgiven by these derivatives at zero inflation to be accurate even up to inflation rates of 30% per year.
20
in computing the effect on the (log of the) aggregate price level of the goods produced by
these firms. We will distinguish between the impact effect on the price level, i.e. the effect
on the moment that the unexpected jump occurs, and the subsequent effects which lead the
price level to adjust up to the full amount δ of the shock.
We can think of the price level just before the shock as the limit: P ≡ limt↑0 P (t; π).
Likewise, we can let the value of the (log of the) price of the aggregate input just before the
shock to be W ≡ limt↑0W (t). Of course, for the price of the aggregate input we have that the
jump is δ = limt↓0W (t) − limt↑0W (t) ≡ limt↓0W (t) − W . As mentioned above, we assume
that d logW (t)/dt = π for t 6= 0. Throughout the exercise the parameters {B,ψ, ρ, σ2} and
the invariant distribution implied by the optimal decision rules, {g, g∗, g} are fixed.
We will denote the price level t periods after the shock P (t; δ, π) for an economy that is
hit by the shock of size δ when the inflation rate of the aggregate input before and after the
cost shock is π. We will let the price level right before the shock to be denoted by P . We
distinguish between the impact effect, which we denote by Θ(δ, π), and the subsequent rate
of change of the (log of the) price level, denoted by θ(t; δ, π). Thus
P (t; δ, π) = P + Θ(δ, π) +
∫ t
0
θ(s; δ, π)ds (7)
Whenever it is clear, we omit δ and π from the expression for P , Θ and θ. Also, while P is
the log of the aggregate price level, whenever it is clear we will refer to it as just the price
level.
Since we are measuring P in logs, then θ(s; δ, π) is the inflation rate of the CPI s periods
after the shock δ has occurred in an economy with steady state inflation rate π. In particular,
after the impact effect at time t = 0, the term θ(s; δ, π)ds gives the contribution to the
average (log) price of the firms that are adjusting prices at times between s and s + ds.
This contribution is equal to θ(s; δ, π) =[∆+p λ
+a (s)−∆−p λ
−a (s)
], where ∆+
p and ∆−p are the
same as in steady state, but the frequency of price increases and decreases, λ+a (s) and λ−a (s),
are time varying. The reason these frequencies are time varying is that the density of the
21
distribution of firms indexed by their price gap g, denoted also by f(g, t), is time varying.
This density changes through time because the cost shock δ and the price changes that occur
right after the cost shock have displace it from its steady state. Thus, even following the
same time invariant decision rules as in steady state, it takes time for the density to return
to its steady state level described by equation (6). See the Appendix C for the discrete time
and discrete state space analog computation of the path of the distribution and of λ’s, or in
Alvarez and Lippi (2019) for the continuous time characterization of the impulse response
using an eigenvalue-eigenfunction decomposition.
We will compare the path of the log of the aggregate price level P (t; δ, π) against the path
for the price level of the aggregate input. Recall that the aggregate input growths at rate π
before and after the shock, and jumps by δ log points at time t = 0. In the long run prices
will increase by as much as the shock to the aggregate input, so that
δ = limt→∞
[P (t; δ, π)− πt]− P = limt→∞
[W (t)− πt]− W (8)
Impact effect. We define the impact effect Θ as the jump in the price level at t = 0, i.e.
Θ(δ, π) = limt↓0
P (t; δ, π)− P (9)
To be clear, if prices will be fully flexible, we will have P (t) = W (t) + µ for some constant µ
at all times. With menu costs, after the jump in the aggregate input prices, we expect that
Θ ≤ δ and that overt time the price level P (t) catches up with the increases in the path of
W (t). Later we show that this is true for values of δ and π/σ2 that are not too large.
The impact effect is simple to compute following this two steps procedure. First we shift
the distribution of price gaps from the steady state to the one right after the shock, but
before the prices adjust, so the new density is f(g+ δ) with support [g− δ, g− δ]. This is so
because with the common increase in cost, the price gap of each firm decreases by δ. Second,
using the lack of first order strategic complementarity, all the firms that end up with price
22
gaps g below g will increase their prices from their new value for g to g∗. Thus:
Θ(δ, π) =
∫ g
g−δ(g∗ − g) f (g + δ) dg (10)
This gives the following derivatives:
∂
∂δΘ(δ, π) =
(g∗ − g + δ
)f(g)
+
∫ g
g−δ(g∗ − g) f ′ (g + δ) dg
∂2
∂δ2Θ(δ, π) = f
(g)
+(g∗ − g + δ
)f ′(g)
+
∫ g
g−δ(g∗ − g) f ′′ (g + δ) dg
Evaluating these expressions at δ = 0, and using that, by definition Θ(0, π) = 0, and that at
the exit points of the invariant density we have f(g) = 0, we obtain the following expansion
of Θ on δ:
Θ(δ, π) =1
2∆+p f′(g) δ2 + o(δ2) (11)
As argued elsewhere, see Alvarez and Lippi (2014), Alvarez, Le Bihan, and Lippi (2016) and
Alvarez, Lippi, and Passadore (2016), if the shock δ is small the impact effect Θ is very small,
mathematically speaking Θ is of second order in δ. We can also see that the leading coefficient
of δ increases with inflation since, as explained above, as π/σ2 increases, the density becomes
more concave in the lower segment, until in the limit f ′(g) → ∞ as π/σ2 → ∞. Thus the
impact effect is of smaller order than δ, but the coefficient of δ2 increases with inflation.
Whether this is an important effect for the level of inflation rates for the period of Argentina
under consideration is an important issue that we will discuss below.
Two extreme examples help to organize ideas. First, consider the case where π/σ2 = 0,
then f ′(g) = 1/(∆+p )2 is constant, and thus for δ ≤ ∆+
p , equation (10) becomes
Θ(δ, π) =1
(∆+p )2
∫ g
g−δ(g∗ − g)
(g − g + δ
)dg =
δ2
2
1
∆+p
(1 +
1
3
δ
∆+p
)
23
As in the general case, for small δ, the value of Θ is very small. Yet when δ is large, say
in the order of magnitude of ∆+p , the impact effect can be large. For instance, if δ = ∆+
p
we have Θ(δ, π) = 23δ which is smaller than δ, but of the same order of magnitude than δ.
Second, consider the other extreme case, where π/σ2 → ∞. In this case f converges to a
uniform distribution between [g, g∗] and thus equation (10) becomes
Θ(δ, π) =1
∆+p
∫ g
g−δ(g∗ − g) dg = δ
(1 +
δ
2
1
∆+p
)
which is of order δ. Note that for small δ this gives the same answer as the case of full price
flexibility, i.e. Θ ≈ δ, and indeed it converges to a version of Caplin and Spulber’s (1987)
neutrality case. Interestingly, when δ is not infinitesimal, then Θ > δ. In this case, since
P (t)− P −πt→ δ as t→∞, there must be an overshooting in the short run, and thus prices
should have an eco and oscillate as they converge to their path. Indeed if δ = ∆+a we have
Θ(δ, π) = 32δ > δ.
The stark difference between the cases with π/σ2 ≈ 0 and π/σ2 → ∞ calls for an
evaluation in the case of Argentina during the period of interest where inflation rate is quite
high, but far away from hyper-inflationary levels, say on the order of 25% per year, when
we exclude the peaks. Is this closer to the π/σ2 → ∞ limit, or is it closer to the π/σ2 → 0
limit. Additionally, are the size of the cost changes for δ for this period in Argentina large
enough that we have to go beyond the approximation in equation (11)? Note that a relevant
theoretical comparison is how large is δ relative to ∆+p . Motivated by these considerations
we will evaluate the relevant expressions for calibrated parameters values and compare them
with the “observed” impact effects.
Initial slope of the impulse response. We now characterize the initial inflation rate,
just after the impact effect. For this we consider a very short time interval right after the
shock, which we denote by ∆. On the one hand, we note that immediately after the shock,
there is no density near the upper bound g. On the other hand, there is a strictly positive
24
density at the lower bound g. Recall that price increases will occur for the firms in the lower
bound of the inaction region which get an idiosyncratic increase in cost, and hence a decrease
in g. Using the assumption of the brownian motion for the idiosyncratic shocks, it can be
shown that about 1/2 the firms at the lower bound will increase prices in the very small
interval of time following the aggregate cost shock. This means that for a very short interval
immediately after the impact effect there is an extremely large number of price increases,
and almost no price decreases.
In particular, after the shift due to the common cost shock, the density is zero in the
upper interval g ∈ [g − δ, g]. Thus, λ−a (∆) → 0 as ∆ → 0. On the other hand, after the
impact effect, and differently from what happens at steady state, there is a positive density
at g = g. This density is equal to 0 < f(g + δ) < f ′(g)δ, where the inequality holds due
to the concavity of the steady state density f in [g, g∗]. Consider the discrete time discrete
state approximation developed in Appendix C where each step of the process for g and of
the discretized steady state distribution are of size√
∆σ. The number of firms changing
prices per unit of time λ+a (∆) equals the density at the boundary times the step size
√∆σ
times the probability that those firms have an increase in cost, denoted by pd, divided by
the length of the time period ∆. For a diffusion, as ∆ → 0 then pd → 1/2. Thus the
fraction of firms changing prices per unit of time is f(g + δ)σ/(
2√
∆)
. This implies that
λ+a (∆)→∞ as ∆→ 0. We have then θ(∆)→ λ+
a (∆)∆+p →∞ as ∆→ 0. Yet, θ(∆)∆→ 0,
as ∆ → 0, so the integral for P (t) is still well defined. An alternative more general way to
show that the slope of the impulse response is infinite is in Alvarez and Lippi (2019) using a
eigenvalue-eigenfunction decomposition of the relevant linear operator.
3.5 Comparative static of cost shocks
In this section we compare the effect of a small and a large cost shock, say δ = 0.01 and
δ = 0.1 for three economies: a low inflation one π = 0.025, a large inflation one, π = 0.25
and one close to the hyperinflationary range, π = 2.5. Recall that cost shocks are measured
25
in logs, so we are trying 1% and 10% once a for all shocks. Inflation rates are measured as
annually continuously compounded, so we are trying 2.5%, 25% and 250%, but in the last two
cases recall that continuously compounded and annually compounded can be meaningfully
different.13
Calibration. We use the same parameters for all cases. The parameters for the firm
problem are chosen so that at π = 0.25, i.e. 25% annual continuously compounded inflation,
the steady state statistics resemble the same statistics in Argentina for the period under
study. We use η = 7, which has a markup of just above 15% and a fixed cost of ψ = 0.012
yearly frictionless profits. We use an annual discount rate ρ = 0.04 and an annual volatility
of idiosyncratic shock of σ = 0.20.
With these parameters the model implies ∆+p = 0.12 and ∆−p = 0.097. The average
number of price changes per year are λ+a = 2.88 and λ−a = 0.89.14 These figures are similar
to the averages for Argentina, when we omit periods of abnormal cost increases. The size
of cost changes in the data is around 10%–11% (see figure 8). The annual number of price
changes measured as the fraction of outlets changing prices in a “normal” month times 12
are λ+a ≈ 0.24× 12 = 2.88 and λ−a ≈ 0.073× 12 = 0.88 (see figure 7).
We will discuss the effect of small (δ = 0.01 or 1%) and large (δ = 0.10 or 10%) cost
shocks for each of the three continuously compounded annualized inflation rates we consider:
π = 0.025, π = 0.25 and π = 2.5. The plots corresponding to each of the inflation rates are
in figure 3, figure 4, and figure 5, respectively. The small shock is of a size we consider in the
upper bound of what a normal monetary shock is. The large cost shocks are similar of the
type of shocks we argue had occurred in Argentina during the period under consideration,
which we view as extremely large. The three inflation rates correspond to: (i) a “common”
inflation rate for a developed economy (π = 0.025 or 2.5% c.c. per year), (ii) a high inflation
13Continuously compounded yearly inflation at rate π implies that the ratio of prices (or cost) at the endof the year relative to prices at the beginning of the year is eπ. For example, with π = 0.25, this ratio ise0.25 ≈ 0.28, and for π = 2.5, this ratio is e2.5 ≈ 12.2!
14We use a time period ∆ = 1/365/12, so two hours.
26
rate which is about the average running inflation during the period of study for Argentina
when we exclude the spikes we associate to the jumps in cost (π = 0.25 or 25% c.c. per
year), and (iii) a very large inflation on the hyperinflation range of almost 23% monthly
compounded inflation (π = 2.5 or 250% c.c. per year).15 For each of the inflation rates we
present three panel of plots: the first for the path of the log of the aggregate CPI level and
for the path of the log of the price of the aggregate input, the second panel with the density
of the invariant distribution right before and right after the cost shock, and the third panel
with the path for the (monthly moving average of the) frequency of price increases and price
decreases. Each panel displays two cases, corresponding to the small cost change (1%) in the
left side of the panel, and corresponding to the large cost change (10%) in the right side of
the panel. In total there are nine subplots for each of the three inflation rates.
Some general comments on the objects of figure 3, figure 4, and figure 5 are in order.
First, in the top panel for each figure we have the path of the log of the nominal price for the
CPI and of the path of the log of the nominal price of the aggregate input. We normalize the
price of the nominal input so that at the time of the cost shock, both the core CPI and the
price of the aggregate input are equal. The time of the cost shock is labeled t = 1, and time is
measured in years for the first and third panels. Second, in the middle panel for each figure
we have the invariant distribution for the price gaps at steady state and its version right
after the cost shock, but before the price changes. In the horizontal axis we have indicated
the values of g, g∗, g. The price gap is measured in log point deviations, so that g = 0.1
represent 10% log point difference between the static maximizing markup and the markup
corresponding to that value of g. We have shaded in red the distribution of price gaps right
after the cost shocks. This shaded area measures the fraction of firms (or products) that
change prices on impact, i.e. at the time of the cost shock. Third, the bottom panel displays
the average number of price changes per year. This is defined as follows: we take the fraction
of firms (or products) that change prices per model period and then we divided it by the
15To be clear, for π = 2.5 annual continuously compounded rate, we have that the monthly compoundedrate is
(e2.5/12 − 1
)× 100 ≈ 23.
27
Figure 3: Pass-through of nominal cost shocks for low inflation (π = 2.5% c.c.)
(a) (b)
(c) (d)
(e) (f)
28
length of the model period. In the plot we display a centered monthly moving average of this
number.16 We take a monthly moving average for comparability with the Argentine data,
which uses monthly frequency, and also to smooth out the large jump. Note that at the time
of the increase of the cost, these frequencies increase smoothly for about a month due to the
fact that we use a centered moving average.
We finish this section with a brief discussion on how the pattern of the different statistics
in these figures illustrates the analytical properties derived above. First we discuss the
difference between the impact effect of the jump in the cost of the aggregate input on the
price level, as well as rate of converge of the price to the cost. Let’s first concentrate on the
case of low (π = 0.025) and high inflation (π = 0.25), i.e. figure 3 and figure 4 respectively.
Even though the inflation rate that roughly corresponds to Argentina is large case (π = 25%
c.c. per year), the pattern for the pass through is similar in figure 3 and figure 4. For both
inflation rates, the instantaneous pass-though is larger for the large cost shock (as can be
seen comparing the left and right subplots). Nevertheless, as expected, in the case of high
inflation (π = 25%) the pass-through is higher and the convergence is faster, i.e. the half
life of the shock with high inflation is one half relative to low inflation one. The convergence
rates in the figures with large shocks are so high, with half-lives below two months, that we
are very close to full price flexibility.
The case of very large inflation of figure 5 with a continuously compounded inflation
π = 250% per year is different. As anticipated in the theoretical section, large shocks in
an inflationary economy induce an overshooting of the price level on impact as shown in
figure 5b. As inflation is very high, firms that adjust prices (and paid the menu cost) find
it optimal to save on future menu costs by raising prices by more than the cost shock. The
case of a small shock depicted in figure 5a instead, is similar to Caplin and Spulber (1987),
also as expected.
16The centered moving average at time t takes an average of half of (1/∆)/12 model periods before thedate t, and half of (1/∆)/12 model periods after the date t, where ∆ is the length –measured in years– ofthe model period. See section C for details on the computations.
29
Figure 4: Pass-through of nominal cost shocks for high inflation (π = 25% c.c)
(a) (b)
(c) (d)
(e) (f)
30
Figure 5: Pass-through of nominal cost shocks for very high inflation (π = 250% c.c.)
(a) (b)
(c) (d)
(e) (f)
31
Now we turn to the middle panel of each of the three figures, which itself is useful to
understand the instantaneous pass-through of the top panel just discussed. Note that for
low inflation (π = 2.5% c.c. per year) the invariant distribution is almost a tent-map, as it
should be for exactly zero inflation. Indeed it is known theoretically, that the effect of steady
state inflation around zero inflation is very small. For high inflation (π = 25% c.c. per year)
the invariant distribution is convex-concave, as explained in the theoretical section above.
Also, as explained in the theory section, comparing the small (δ = 0.01) and large cost shocks
(δ = 01) corresponding to the left and right panels respectively, it is seen that the number of
firms (or products) that change prices (i.e. the size of the red shaded area) increases more
than proportionally as the shock increases from 1% to 10%. Alternatively, we can see that
the approximation that the impact effect on prices is proportional to the square of the cost
shock δ is accurate for this range of shocks. Moreover, the size of the red shaded area for the
10% shock in the low inflation rate π = 2.5% case is smaller than in the high inflation rate
π = 25% case, due to the convex-concave nature of the invariant distribution for the higher
inflation rates. Again, the case of very large inflation (π = 250% c.c. per year) is different.
The share of firms changing prices in impact is much closer to be proportional to δ than in
the cases of lower inflation, as can be seen in figure 5d.
Lastly we turn to the behavior of the frequency of price increases and decreases, the
bottom panel of the figures for each inflation rate. First, note that that for low inflation
(π = 2.5% c.c. per year or figure 3) while the rise on the frequency of price increases is
moderately larger than the decline on the frequency of price decreases for small cost shocks
(δ = 0.01 or left panel), this difference is much larger for the case of high cost changes
(δ = .10 or right panel). The pattern is similar in the case of high inflation (π = 25% c.c.
per year, or figure 4), except that the differences between increase and decreases are a bit
more stark. Instead, again, the situation for vey large inflation (π = 250% c.c. per year, or
figure 5) is different. Since there are almost no price decreases, there is no detectable change
on them. On the expected number of price decreases, the behavior is very different from
32
small cost shocks (δ = 0.01 or left panel) than from large cost shocks (δ = 0.1 or right panel).
For small cost shocks we have the one time blip that is characteristic of the mechanism in
Caplin and Spulber (1987). Instead, as explained in the theory above, for large cost shocks,
there is overshooting which leads to a subsequent echo effect, which is seen in the damped
oscillations in the path of the frequency of price increases.
4 Argentina’s Evidence on Large Cost Shocks and Price
Dynamics
In the previous section we studied how firms facing menu costs of price adjustments react
to unexpected shocks to nominal marginal costs. We then studied the aggregate behavior
of prices paying particular attention to the effect of these shocks on the average price level,
on the size of price changes, and on the frequency of price adjustment, both on impact and
over time. In this section we look at how prices in the City of Buenos Aires reacted to the
large nominal shocks described in section 2 and draw conclusions on the applicability of the
model.
Figure 6 provides an overview of the behavior of our proxy of the nominal cost and of
core prices. We prefer to look at core inflation because as it excludes seasonal and regulated
goods, this measure of prices avoids the mechanical direct impact that changes in regulated
prices have on the CPI.
There are three large jumps in costs: one in early 2014, a second one in early 2016, and a
third one in May-September 2018. The first one is mainly due to a 23% devaluation that took
place in the second half of January. As our measure of costs is based on monthly averages,
our cost proxy jumps in December 2013 and in January 2014. The 40% weight of tradable
goods in our cost measure implies that the jump in cost is slightly above 9%, roughly in sink
with the size of price changes in the data (see figure 8) and in the simulated examples in
section 3.5. Figure 6 also shows that there is a spike in inflation associated to each spike in
nominal costs. Between November and February 2013 the cost proxy increased 12% while
33
Figure 6: Nominal costs and price levels
Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan190
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Lo
ga
rith
mic
po
ints
0
0.02
0.04
0.06
0.08
0.1
0.12
Mo
nth
ly lo
ga
rith
mic
diffe
ren
ce
Core Price
Cost
Core inflation
Cost inflation
Note: Cost is a weighted geometric average of regulated prices, the exchangerate and wages with weights 0.1, 0.4 and 0.5, respectively.Source: Prices are from for the City of Buenos Aires Statistical Office. Ex-change rates from the central bank and wages from Ministerio de Trabajo(2018).
the price level increased by 8%. The second shock took place in the first half of 2016. It
consisted of a sequence of cost shocks stemming from the impact of the removal of capital
controls on the exchange rate, and from the change in the relative price of regulated energy
prices as shown in table 1. The impact effect relative to the size of the shock is smaller than
in the first shock. The persistence of the shocks is reflected in the persistence of the high
inflation. Finally, in the first quarter of 2018 there are nominal shocks of about 4% related
to regulated prices and starting in May there are two exchange rate lead spikes in nominal
costs with peaks of 7% in May and 12% in September.
Several issues prevent us from using the data underlying figure 6 to estimate the impulse
response of core prices to cost shocks analyzed in section 3.4. (i) At the time a shock hits prices
might still be adjusting to previous shocks. (ii) The cost shock might have been partially
34
anticipated.17 (iii) An aggregate shock might change the relative price between consumer
goods and wholesale goods. (iv) There might be other aggregate cost or productivity shocks
not captured by our cost proxy. Nevertheless, we can check if the frequency and if the size
of price changes in the data are consistent with the theoretical analysis in 3, illustrated in
figures 3 to 5.
Figures 7 and 8 show the frequency and the size of price changes. The figures are based
on the data underlying the City of Buenos Aires core consumer price index (IPCBA-resto).
The city collects approximately 70,000 prices per month for 628 goods and services.18 The
frequency of price changes is computed as the fraction of prices that either increased or
decreased between two consecutive observations within the goods and services included in
the core measure of inflation. The size of price changes is the geometric equally weighted
average of the absolute value of price increases/decreases. The methodology for computing
these statistics is described in Alvarez et al. (2019) where we discuss the property of this
simple estimator and perform robustness checks.
Figure 7 shows the fraction of outlets changing price each month, our proxy for costs and
the core inflation level. Observe first, that the average level of the fraction of price increases
in “normal” times is 0.24 and the one for price decreases is 0.073. The magnitude of the
frequency of price decreases is interesting because it indicates that for the levels of underlying
inflation during 2012-18 in Argentina the benchmark menu cost model of Sheshinski and Weiss
(1979) is unlikely to be the appropriate one–something that we will also see as we examine
other statistics. It rules out the case of π/σ → ∞ in figure 5, pointing out the importance
of idiosyncratic shocks that induce some firms to lower prices even when costs are rising at
a cruising speed of 2% per month.
Second, the three cost shocks identified in our narrative in 2014, 2016, and 2018 are large
in the sense that their size is similar to the size of the price changes in normal times. In the
17 This issue is especially relevant for the devaluation of December 2015. See for example Neumeyer (2015)and Levy-Yeyati (2015)
18 The instructions to the enumerators and other methodological issues are described in Direccion Generalde Estadistica y Censos (2013).
35
Figure 7: Cost shocks and the frequency of price changes
Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan190
0.1
0.2
0.3
0.4
0.5
0.6F
ractio
n o
f o
utle
ts c
ha
ng
ing
price
s
-0.05
0
0.05
0.1
0.15
0.2
0.25
Mo
nth
ly lo
ga
rith
mic
diffe
ren
ce
Note: λ+ is the fraction of outlets increasing prices each month. λ− is the analogue for outletsdecreasing prices. Cost is a weighted geometric average of regulated prices, the exchange rateand wages with weights 0.1, 0.4 and 0.5, respectively. The doted line is the core inflation forthe City of Buenos Aires (IPCBA - resto) .Source: Prices are from for the City of Buenos Aires Statistical Office. Exchange rates fromthe central bank . Wages from Ministerio de Trabajo (2018).
three cases, the reaction of the frequencies of price changes is consistent with the predictions
of the model for the case of a large shock with high inflation, illustrated in figure 4f. The
shock at the beginning of 2014 is short lived and of a similar magnitude to the δ = 0.1 in
the simulated example. As suggested by the theory, the data shows a contemporaneous spike
of the fraction of outlets raising prices to 0.45. The transitory decrease in the fraction of
outlets lowering prices seems to be in the data but it is hard to distinguish it from noise.
The second episode corresponds to the sequence of cost shocks that took place in the first
half of 2016. These shocks also induced an increase in the fraction of outlets raising prices
that is less pronounced and more persistent than the one in 2014. The frequency of price
increases peaks at 0.38 in January 2016 and at 0.35 in May 2016. There seems to be a fall
36
in the fraction of outlets lowering prices in the period leading to the devaluation but, again,
it is hard to distinguish it from noise in the data. Finally, the fraction of outlets raising
prices rises throughout 2018 reacts to the cost shocks, with peaks of 0.43 in June and 0.57
in September, coinciding with important jumps in the exchange rate.
Figure 8: Cost shocks and the Size of Price Changes
Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan190.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Absolu
te v
alu
e o
f %
change in p
rices
-0.05
0
0.05
0.1
0.15
0.2
0.25
Month
ly logarith
mic
diffe
rence
Note: ∆+ is the average percentage change in prices across outlets conditional on a priceincrease. ∆− is the analogue for price decreases. Cost is a weighted geometric average ofregulated prices, the exchange rate and wages with weights 0.1, 0.4 and 0.5, respectively.The doted line is the core inflation for the City of Buenos Aires (IPCBA - resto) .Source: Prices are from for the City of Buenos Aires Statistical Office. Exchange rates fromthe central bank . Wages from Ministerio de Trabajo (2018).
Figure 8 describes the absolute size of price changes conditional on a price change taking
place. In “normal” times the absolute size of positive and negative price changes are similar,
around 10-11%. This is consistent with the theory. However, the fact that the absolute
value of price decreases is sometimes higher than the one for price increases after 2014 is
inconsistent with the theory.19
19These differences may just be noise.
37
The reaction of the size of price changes to the cost shocks also gives empirical support
to the menu cost model. In the theory, when there is a large cost shock many firms end
up with a markup that is outside the inaction range as shown is figure 4d. This implies
that the magnitude of price increases has to increase in the presence of large unanticipated
shocks. The magnitude of price decreases on the other hand, should not change as a result of
increases in nominal marginal costs as no firm’s markup are pushed above the upper band,
g. These two predictions of the theory are observed in the data. For the three shocks, we
observe significant increases in the magnitude of price increases in figure 8, which reach 50%
in 2018 and 2018. Also, the magnitude of price decreases does not show abnormal patterns
around the time of cost shocks.
We conclude this section by saying that the behavior of the frequency and the size of
price changes supports the pass-through of costs shocks to prices prices predicted by the
menu cost model of price adjustment. In “normal” times with small shocks, the frequency
and the size of price changes are consistent with the simulations for π = 0.25 and δ = 0.01
in section 3.5, supporting the choice of parameter values for ψ/B and π/σ used in the
simulations. The frequency of price decreases of 0.073 and the fact that the size of price
increases and decreases implies that idiosyncratic firm shocks are important. The expected
duration of prices, computed as (λ+ + λ−)−1, is 3.2 months supporting the importance of
price rigidities. For the episodes that we interpret as large unanticipated cost shocks, the
frequency and the size of price adjustment react as the theory predicts with large increases
in the size and in the frequency of price increases and with no discernible effects on these
variables for the case of price decreases. This leads us to conjecture that the fast pass-through
of costs to prices, predicted by the theory but hard to estimate in our dataset, is likely to be
present in the data.
38
References
Alvarez, Fernando, Martin Gonzalez-Rozada, Andres Neumeyer, and Martin Beraja. 2019.“From Hyperinflation to Stable Prices: Argentina’s evidence on menu cost models.” Quar-terly Journal of Economics 134 (1):452–505.
Alvarez, Fernando and Francesco Lippi. 2019. “The Analytic Theory of a Monetary Shock.”Tech. rep., The University of Chicago.
Alvarez, Fernando, Francesco Lippi, and Juan Passadore. 2016. “Are State and Time depen-dent models really different?” NBER Macroeconomics Annual 2016, Volume 31 .
Alvarez, Fernando E., Herve Le Bihan, and Francesco Lippi. 2016. “The real effects ofmonetary shocks in sticky price models: a sufficient statistic approach.” The AmericanEconomic Review 106 (10):2817–2851.
Alvarez, Fernando E. and Francesco Lippi. 2014. “Price setting with menu costs for multiproduct firms.” Econometrica 82 (1):89–135.
Bassetto, Marco and Christopher Phelan. 2015. “Speculative runs on interest rate pegs.”Journal of Monetary Economics 73 (C):99–114.
Bonadio, Barthlmy, Andreas Fischer, and Philip Saur. 2016. “The speed of exchange ratepass-through.” Globalization Institute Working Papers 282, Federal Reserve Bank of Dal-las.
Caplin, Andrew S. and Daniel F. Spulber. 1987. “Menu Costs and the Neutrality of Money*.”The Quarterly Journal of Economics 102 (4):703–725.
Cavallo, Alberto. 2013. “Online and Official Price Indexes: Measuring Argentina’s Inflation.”Journal of Monetary Economics 60 (2):152–165.
Direccion General de Estadistica y Censos, CABA. 2013. “Indice de Precios al Consumidorde la Ciudad de Buenos Aires. Principales aspectos metodolgicos.”
Gagnon, Etienne. 2009. “Price Setting During Low and High Inflation: Evidence fromMexico.” Quarterly Journal of Economics 124 (3):1221–1263.
Gagnon, Etienne, David Lopez-Salido, and Nicolas Vincent. 2013. “Individual Price Ad-justment along the Extensive Margin.” In NBER Macroeconomics Annual 2012, Volume27, edited by Daron Acemoglu, Jonathan Parker, and Michael Woodford. University ofChicago Press, 235–281.
Golosov, Mikhail and Robert E. Jr. Lucas. 2007. “Menu Costs and Phillips Curves.” Journalof Political Economy 115:171–199.
Jones, Charles I. 2011. “Intermediate Goods and Weak Links in the Theory of EconomicDevelopment.” American Economic Journal: Macroeconomics 3 (2):1–28.
39
Karadi, Peter and Adam Reiff. 2018. “Menu Costs, Aggregate Fluctuations, and LargeShocks.” American Economic Journal: Macroeconomics forthcomming.
Kehoe, Patrick and Virgiliu Midrigan. 2015. “Prices are sticky after all.” Journal of MonetaryEconomics 75:35 – 53.
Levy-Yeyati, Eduardo. 2015. “De que hablamos cuando hablamos de dolar e inflacion?” BlogFoco Economico URL http://focoeconomico.org.
Ministerio de Trabajo, Empleo y Seguridad Social. 2018. “Boletn de remuneraciones de lostrabajadores registrados.”
Neumeyer, Andres. 2015. “El dolar y los precios: anticipando el fin del cepo.” Blog FocoEconomico URL http://focoeconomico.org.
Sheshinski, Eytan and Yoram Weiss. 1979. “Demand for Fixed Factors, Inflation and Ad-justment Costs.” Review of Economic Studies 46 (1):31–45.
40
A Derivation of the profit function
Define
F (g, x, z) = Π(g)Ax1−ηz where Π(g) =[eg+m − 1
]e−η(g+m)
Thus
Π′(g) = e(1−η)(g+m) (1− η) + ηe−η(g+m) and evaluating it at g = 0
Π′(0) = e−ηm [em (1− η) + η] =⇒ em =η
η − 1
Likewise
Π′′(g) = e(1−η)(g+m) (1− η)2 − η2e−η(g+m) and evaluating it at g = 0
Π′′(0) = e(1−η)m (1− η)2 − η2e−ηm =
(η
η − 1
)1−η
(1− η)2 − η2
(η
η − 1
)−η=
(η
η − 1
)−η [(η
η − 1
)(1− η)2 − η2
]=
(η
η − 1
)−η [η (η − 1)− η2
]= −
(η
η − 1
)−ηη = −
(η − 1
η
)ηη
The level of the optimized value of Π is:
Π(0) = [em − 1] e−ηm = e(1−η)m − e−ηm =
(η
η − 1
)(1−η)
−(
η
η − 1
)−η=
(η
η − 1
)−η [η
η − 1− 1
]=
(η
η − 1
)−η [1
η − 1
]A second order expansion of F (g, x, z) around (0, x, z) gives:
F (g, x, z) = Π′(0)Ax1−ηg + Π(0)(1− η)Ax−ηz(x− x) + Π(0)Ax1−η(z − z)
+1
2Π′′(0)Ax1−ηzg2 +
1
2Π(0)A(1− η)(−η)x−η−1z(x− x)2 + 0
1
2(z − z)2
+ Π′(0)(1− η)Ax−ηz(x− x)g + Π′(0)Ax1−η0(z − z)g + Π(0)Ax1−η0(z − z)(x− x)
+ o(||(x− x, g)||2
)Since Π′(0) = 0 we have:
F (g, x, z) = Π(0)(1− η)Ax−η(x− x) +1
2Π′′(0)Ax1−ηg2 + Π(0)Ax1−η(z − z)
+1
2Π(0)A(1− η)(−η)x−η−1(x− x)2 + o
(||(x− x, g)||2
)
41
Finally, ignoring the terms that are smaller than second order, or that do not involve g, anddividing by F (0, x, z), the maximum profit at x, we have:
F (g) = −1
2η(η − 1)g2 ≡ Bg2 .
B Value function and optimal decision rules
The value function V (·) and the optimal decision rules {g, g∗, g} solves the following systemof o.d.e. and boundary conditions:
ρV (g) = F (g)− πV ′(g) +σ2
2V ′′(g) for g ∈ [g, g]
as well as value matching and smooth pasting at the boundary of the range of inaction aswell as optimality of g∗:
V (g) = V (g∗)− ψ , V (g) = V (g∗)− ψ , and V ′(g) = V ′(g) = V ′(g∗) = 0
As explained in Alvarez et al. (2019), the decision rules depend on ψ/B, π/σ2 and ρ/σ2.One can further simplify the problem by considering the case in which ρ → 0, the so called“ergodic” control case. Indeed the solutions are very insensitive to ρ when it takes smallvalues –one can show that the derivative of them with respect to ρ is zero when evaluatedat ρ = 0. The appendix in Alvarez et al. (2019) describes the limit that defines the ergodiccontrol case and characterizes its solutions. One can also compare the solutions obtainednumerically with the method described in Appendix C, which requires ρ > 0; they arealmost identical. Hence, to simplify, we will parametrized the solutions by two ratios: ψ/Band π/σ2.
C Numerical solution of value function and simulated
statistics
An alternative way to compute both the value function and optimal policies, as well as theeffect of cost shocks of the price level is to discretize time as well as the state space. Weconsider there a discrete time and discrete state version of the problem. We let ∆ be thelength of the time period. We will let
√∆σ the distance between any two points in the state
space. For this we represent the Brownian motion during the range of inaction as:
g(t+ ∆)− g(t) =
+σ√
∆ with probability pu = 12
(1− π
√∆σ
)−σ√
∆ with probability pd = 12
(1 + π
√∆σ
)so that E [g(t+ ∆)− g(t)] = −π∆ and E [g(t+ ∆)− g(t)]2 = ∆σ2. Thus we can let the statespace be G = {k
√∆σ}k=M
k=−M with typical element gk ∈ G. The value function is just a vector
42
V ∈ R2M+1. The discretized version solves the following Bellman equation:
Vk = max
{ψ + max
jVj , ∆F (gk) +
1
1 + ∆ρ
[puVmax{k+1,K} + pdVmin{k−1,−K}
]}for k = −K,−K + 1, . . . , K − 1, K. The choice of K should be large enough so the rangeon inaction is well in the interior of G. The choice of ∆ should be small enough so thatit approximates well the continuous time limit. The solution of this problem will give thethresholds g and g, as well as k∗ = arg maxk{Vk} so that g∗ = gk∗ .
We let f be a 2K + 1 positive vector containing the fraction of firms with different pricegaps. For instance fk will be the fraction of firms with price gap gk. The law of motion for thevector with the distribution of firms f(t+1) can be represented by a square stochastic matrixL of 2K+1×2K+1 dimensions, so f(t+∆) = Lf(t), where we use f(t) for the column vectorof 2K + 1 values of the fractions of firms fk(t) a time t with price gap gk. The matrix L canbe thought as the sum of two matrices. The first matrix has zeros in all the entries, exceptin the entires next to the diagonal where it has ether pu or pd, keeping tract of the mass offirms within the inaction region. The second matrix is also sparse, it has zeros and ones, withones indicating that the firms that are outside the range of inaction will transit to optimalreturn point k∗ with probability one. Recall that we have: fj(t + ∆) =
∑k=−K,K Lj,kfk(t).
Let k, k∗ and k the indices of the elements of f that correspond to g, g∗ and g respectively.The elements Li,j are zero, except in the following cases.
1. For k = −K, . . . , k − 1, then Lk∗,k = 1,
2. For k = k + 1, . . . , K, then Lk∗,k = 1,
3. For k = k + 1, . . . , k − 1, then Lk,k+1 = pd and Lk,k−1 = pu,
4. Lk,k+1 = pd and Lk,k−1 = pu, and
5. Lk∗,k = pd and Lk∗,k = pu.
As it is well known the invariant distribution can be obtained by computing the powers ofLT for large value of T . Alternatively, the invariant distribution is the eigenvector associatedwith the eigenvalue equal to one of matrix L.
The fraction of price changes per period of length ∆ is given a λ+a (t) = 1
∆l+ f(t) where
l+ is a vector that adds the components of the vector f(t) with values below g. In particularthe row vector l+ has:
1. l+k = 1 for k = −K, . . . , k − 1,
2. l+k = pd, and
3. l+k = 0 for k = k + 1, . . . , K.
We can define λ−a (t) = 1∆l− f(t) in an analogous way. The row vector l− has:
1. l−k = 1 for k = k + 1, . . . , K,
43
2. l−k
= pu, and
3. l−k = 0 for k = −K, . . . , k − 1.
Thus, the fraction of firms that change prices between t and t + ∆ per unit of time isλa(t) = λ+
a (t) + λ−a (t). Note that the definition of λa, λ+a and λ−a , divides the fraction of
firms that adjust prices by ∆ so that it is comparable, at least for small ∆, to the continuoustime expressions. Hence, if we multiply the expressions for λa(t), λ
+a (t) or λ−a (t) by ∆ we will
obtain fraction of firms that change prices during the interval t and t+ ∆.
Discrete time - discrete state impulse response. We can write the equivalent of theimpulse response on prices using the matrix L. We start with f given by the invariantdistribution implied by L. Then we let f(0; δ) be the 2K + 1 vector of the displaced initialconditions. This is a shift of the mass to the left by an amount equal δ. This vector is givenby:
fk(0; δ) = fk+ν for k = −K, . . . ,K − ν and fk(0; δ) = 0 otherwise, (12)
and where ν = δ/[σ√
∆]
which, for simplicity, we assume to be an integer. Now we can
compute the impact effect Θ:
Θ =
k−1∑k=−K
(gk∗ − gk) fk(0; δ)−K∑
k=k+1
(gk − gk∗) fk(0; δ) , (13)
Then we compute the sequence of distribution of firms indexed by the their price gap as:
f (∆) = [L]j f(0; δ) for j = 1, 2, . . .
With these elements we compute the rate of increase in prices θ’s:
θ(j∆) = ∆+p λ
+a (j∆)−∆−p λ
−a (j∆) for j = 1, 2, . . . where (14)
λ+a (j∆) =
1
∆l+ f(j∆) for j = 1, 2, . . . , and (15)
λ−a (j∆) =1
∆l− f(j∆) for j = 1, 2, . . . . (16)
The analog of the continuous time impulse response is:
P (j∆) = P + π∆ + Θ +
j∑k=1
θ(k∆)∆ for j = 1, 2, . . .
where P is the price level in steady state just before the cost shock, or P = P (−∆).To simplify the exposition we have not taken into account the regular idiosyncratic shocks
that also occur during the period between t = 0 and t = ∆. To correct for this effect we haveincluded π∆ to the initial value of P , which is the correct value since we are starting withthe invariant distribution. This has the interpretation that the shock δ occurs at the end thediscrete period, but before the next period. As ∆→ 0 these effects can be neglected.
44
Documentos de Trabajo
Banco Central de Chile
NÚMEROS ANTERIORES
La serie de Documentos de Trabajo en versión PDF
puede obtenerse gratis en la dirección electrónica:
www.bcentral.cl/esp/estpub/estudios/dtbc.
Existe la posibilidad de solicitar una copia impresa
con un costo de Ch$500 si es dentro de Chile y
US$12 si es fuera de Chile. Las solicitudes se
pueden hacer por fax: +56 2 26702231 o a través del
correo electrónico: [email protected].
Working Papers
Central Bank of Chile
PAST ISSUES
Working Papers in PDF format can be downloaded
free of charge from:
www.bcentral.cl/eng/stdpub/studies/workingpaper.
Printed versions can be ordered individually for
US$12 per copy (for order inside Chile the charge
is Ch$500.) Orders can be placed by fax: +56 2
26702231 or by email: [email protected].
DTBC – 843
The Nonpuzzling Behavior of Median Inflation
Laurence Ball, Sandeeo Mazumder
DTBC – 842
The Propagation of Monetary Policy Shocks in a Heterogeneous Production Economy
Ernesto Pastén, Raphael Schoenle, Michael Weber
DTBC – 841
Índice de sincronía bancaria y ciclos financieros
Juan Francisco Martinez, Daniel Oda
DTBC – 840
The impact of interest rate ceilings on households’ credit access: evidence from a 2013
Chilean legislation
Carlos Madeira
DTBC – 839
On Corporate Borrowing, Credit Spreads and Economic Activity in Emerging
Economies: An Empirical Investigation
Julián Caballero, Andrés Fernández, Jongho Park
DTBC – 838
Adverse selection, loan access and default in the Chilean consumer debt market
Carlos Madeira
DTBC – 837
The Persistent Effect of a Credit Crunch on Output and Productivity: Technical or
Allocative Efficiency?
Patricio Toro
DTBC – 836
Sectoral Transitions Between Formal Wage Jobs in Chile
Rosario Aldunate, Gabriela Contreras, Matías Tapia
DTBC – 835
Misallocation or Misspecification? The Effect of “Average” Distortions on TFP Gains
Estimations
Elías Albagli, Mario Canales, Antonio Martner, Matías Tapia, Juan M. Wlasiuk
DTBC – 834
Forecasting Inflation in a Data-rich Environment: The Benefits of Machine Learning
Methods
Marcelo Medeiros, Gabriel Vasconcelos, Álvaro Veiga, Eduardo Zilberman
DTBC – 833
XMAS: An Extended Model for Analysis and Simulations Benjamín García, Sebastián Guarda, Markus Kirchner, Rodrigo Tranamil
DTBC – 832
Extracting Information of the Economic Activity from Business and Consumer Surveys
in an Emerging Economy (Chile) Camila Figueroa y Michael Pedersen
DTBC – 831
Firm Productivity dynamics and distribution: Evidence for Chile using micro data
from administrative tax records Elías Albagli, Mario Canales, Claudia De la Huerta, Matías Tapia y Juan Marcos Wlasiuk
DTBC – 830
Characterization of the Recent Immigration to Chile Rosario Aldunate, Gabriela Contreras, Claudia De la Huerta y Matías Tapia
DTBC – 829
Shifting Inflation Expectations and Monetary Policy Agustín Arias y Markus Kirchner
DOCUMENTOS DE TRABAJO • Octubre 2019